Some consequences of the classification of finite simple groups.

*(English)*Zbl 0594.20008
Finite groups - coming of age, Proc. CMS Conf., Montreal/Que. 1982, Contemp. Math. 45, 159-173 (1985).

[For the entire collection see Zbl 0565.00006.]

The paper lists various theorems having in common that they were proved by use of the classification of finite simple groups. The results surveyed are arranged in 14 ”examples”, most of them (Exs. 5-11) relating to permutation groups (e.g., the determination of all 2-transitive groups, of which there is no explicit list in this paper). The other examples range from Seitz’s description of subgroups of classical groups containing a maximal torus (Ex. 1), Feit’s result restricting the shape of a Brauer tree (Ex. 2), purely group theoretical properties (e.g., solvability of certain groups, Ex. 3), and number theoretic results (e.g., the fact that certain relative Brauer groups are infinite, Ex. 4) to asymptotics on the number of isomorphism classes of groups of given order (Ex. 12), applications to combinatorial questions (Ex. 13), and the complexity of permutation group algorithms (measured in terms of the lengths of generating permutations, an example being the author’s result that, given a prime p, an element of order p can be found in polynomial time).

The paper lists various theorems having in common that they were proved by use of the classification of finite simple groups. The results surveyed are arranged in 14 ”examples”, most of them (Exs. 5-11) relating to permutation groups (e.g., the determination of all 2-transitive groups, of which there is no explicit list in this paper). The other examples range from Seitz’s description of subgroups of classical groups containing a maximal torus (Ex. 1), Feit’s result restricting the shape of a Brauer tree (Ex. 2), purely group theoretical properties (e.g., solvability of certain groups, Ex. 3), and number theoretic results (e.g., the fact that certain relative Brauer groups are infinite, Ex. 4) to asymptotics on the number of isomorphism classes of groups of given order (Ex. 12), applications to combinatorial questions (Ex. 13), and the complexity of permutation group algorithms (measured in terms of the lengths of generating permutations, an example being the author’s result that, given a prime p, an element of order p can be found in polynomial time).

Reviewer: A.M.Cohen

##### MSC:

20D05 | Finite simple groups and their classification |

20B20 | Multiply transitive finite groups |

20D60 | Arithmetic and combinatorial problems involving abstract finite groups |